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Theorem mexval 35487
Description: The set of expressions, which are pairs whose first element is a typecode, and whose second element is a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mexval.k 𝐾 = (mTC‘𝑇)
mexval.e 𝐸 = (mEx‘𝑇)
mexval.r 𝑅 = (mREx‘𝑇)
Assertion
Ref Expression
mexval 𝐸 = (𝐾 × 𝑅)

Proof of Theorem mexval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 mexval.e . 2 𝐸 = (mEx‘𝑇)
2 fveq2 6907 . . . . . 6 (𝑡 = 𝑇 → (mTC‘𝑡) = (mTC‘𝑇))
3 mexval.k . . . . . 6 𝐾 = (mTC‘𝑇)
42, 3eqtr4di 2793 . . . . 5 (𝑡 = 𝑇 → (mTC‘𝑡) = 𝐾)
5 fveq2 6907 . . . . . 6 (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇))
6 mexval.r . . . . . 6 𝑅 = (mREx‘𝑇)
75, 6eqtr4di 2793 . . . . 5 (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅)
84, 7xpeq12d 5720 . . . 4 (𝑡 = 𝑇 → ((mTC‘𝑡) × (mREx‘𝑡)) = (𝐾 × 𝑅))
9 df-mex 35472 . . . 4 mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡)))
10 fvex 6920 . . . . 5 (mTC‘𝑡) ∈ V
11 fvex 6920 . . . . 5 (mREx‘𝑡) ∈ V
1210, 11xpex 7772 . . . 4 ((mTC‘𝑡) × (mREx‘𝑡)) ∈ V
138, 9, 12fvmpt3i 7021 . . 3 (𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅))
14 xp0 6180 . . . . 5 (𝐾 × ∅) = ∅
1514eqcomi 2744 . . . 4 ∅ = (𝐾 × ∅)
16 fvprc 6899 . . . 4 𝑇 ∈ V → (mEx‘𝑇) = ∅)
17 fvprc 6899 . . . . . 6 𝑇 ∈ V → (mREx‘𝑇) = ∅)
186, 17eqtrid 2787 . . . . 5 𝑇 ∈ V → 𝑅 = ∅)
1918xpeq2d 5719 . . . 4 𝑇 ∈ V → (𝐾 × 𝑅) = (𝐾 × ∅))
2015, 16, 193eqtr4a 2801 . . 3 𝑇 ∈ V → (mEx‘𝑇) = (𝐾 × 𝑅))
2113, 20pm2.61i 182 . 2 (mEx‘𝑇) = (𝐾 × 𝑅)
221, 21eqtri 2763 1 𝐸 = (𝐾 × 𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339   × cxp 5687  cfv 6563  mTCcmtc 35449  mRExcmrex 35451  mExcmex 35452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-mex 35472
This theorem is referenced by:  mexval2  35488  msubff  35515  msubco  35516  msubff1  35541  mvhf  35543  msubvrs  35545
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